A normal mode of a
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
is a pattern of motion in which all parts of the system move
sinusoidal
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
ly with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. These fixed frequencies of the normal modes of a system are known as its
natural frequencies
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...
or
resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions.
The most general motion of a system is a
superposition of its normal modes. The modes are normal in the sense that they can move independently, that is to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
to each other.
General definitions
Mode
In the
wave theory
In historical linguistics, the wave model or wave theory (German ''Wellentheorie'') is a model of language change in which a new language feature (innovation) or a new combination of language features spreads from its region of origin, affecting ...
of physics and engineering, a mode in a
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
is a
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
state of excitation, in which all the components of the system will be affected sinusoidally at a fixed frequency associated with that mode.
Because no real system can perfectly fit under the standing wave framework, the ''mode'' concept is taken as a general characterization of specific states of oscillation, thus treating the dynamic system in a ''linear'' fashion, in which linear
superposition of states can be performed.
Classical examples include
* In a mechanical dynamical system, a vibrating rope is the most clear example of a mode, in which the rope is the medium, the stress on the rope is the excitation, and the displacement of the rope with respect to its static state is the modal variable.
* In an acoustic dynamical system, a single sound pitch is a mode, in which the air is the medium, the sound pressure in the air is the excitation, and the displacement of the air molecules is the modal variable.
* In a structural dynamical system, a high tall building oscillating under its most flexural axis is a mode, in which all the material of the building -under the proper numerical simplifications- is the medium, the seismic/wind/environmental solicitations are the excitations and the displacements are the modal variable.
* In an electrical dynamical system, a resonant cavity made of thin metal walls, enclosing a hollow space, for a particle accelerator is a pure standing wave system, and thus an example of a mode, in which the hollow space of the cavity is the medium, the RF source (a Klystron or another RF source) is the excitation and the electromagnetic field is the modal variable.
* When relating to
music
Music is generally defined as the The arts, art of arranging sound to create some combination of Musical form, form, harmony, melody, rhythm or otherwise Musical expression, expressive content. Exact definition of music, definitions of mu ...
, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "
overtones".
The concept of normal modes also finds application in other dynamical systems, such as
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultra ...
,
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
,
atmospheric dynamics and
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
.
Most dynamical systems can be excited in several modes, possibly simultaneously. Each mode is characterized by one or several frequencies, according to the modal variable field. For example, a vibrating rope in 2D space is defined by a single-frequency (1D axial displacement), but a vibrating rope in 3D space is defined by two frequencies (2D axial displacement).
For a given amplitude on the modal variable, each mode will store a specific amount of energy because of the sinusoidal excitation.
The ''normal'' or ''dominant'' mode of a system with multiple modes will be the mode storing the minimum amount of energy for a given amplitude of the modal variable, or, equivalently, for a given stored amount of energy, the dominant mode will be the mode imposing the maximum amplitude of the modal variable.
Mode numbers
A mode of vibration is characterized by a modal frequency and a mode shape. It is numbered according to the number of half waves in the vibration. For example, if a vibrating beam with both ends pinned displayed a mode shape of half of a sine wave (one peak on the vibrating beam) it would be vibrating in mode 1. If it had a full sine wave (one peak and one trough) it would be vibrating in mode 2.
In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to t ...
, we have a radial coordinate and an angular coordinate. If one measured from the center outward along the radial coordinate one would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because only half of the disk is considered due to the anti-symmetric (also called
skew-symmetry) nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1. So the mode number of the system is 2–1 or 1–2, depending on which coordinate is considered the "first" and which is considered the "second" coordinate (so it is important to always indicate which mode number matches with each coordinate direction).
In linear systems each mode is entirely independent of all other modes. In general all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes.
Nodes
In a one-dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero. These nodes correspond to points in the mode shape where the mode shape is zero. Since the vibration of a system is given by the mode shape multiplied by a time function, the displacement of the node points remain zero at all times.
When expanded to a two dimensional system, these nodes become lines where the displacement is always zero. If you watch the animation above you will see two circles (one about halfway between the edge and center, and the other on the edge itself) and a straight line bisecting the disk, where the displacement is close to zero. In an idealized system these lines equal zero exactly, as shown to the right.
In mechanical systems
Coupled oscillators
Consider two equal bodies (not affected by gravity), each of
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
''m'', attached to three springs, each with
spring constant ''k''. They are attached in the following manner, forming a system that is physically symmetric:
:
where the edge points are fixed and cannot move. We'll use ''x''
1(''t'') to denote the horizontal
displacement of the left mass, and ''x''
2(''t'') to denote the displacement of the right mass.
If one denotes acceleration (the second
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of ''x''(''t'') with respect to time) as
, the
equations of motion are:
:
Since we expect oscillatory motion of a normal mode (where ω is the same for both masses), we try:
:
Substituting these into the equations of motion gives us:
:
Since the exponential factor is common to all terms, we omit it and simplify:
:
And in
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
representation:
:
If the matrix on the left is invertible, the unique solution is the trivial solution (''A''
1, ''A''
2) = (''x''
1, ''x''
2) = (0,0). The non trivial solutions are to be found for those values of ω whereby the matrix on the left is
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar ...
i.e. is not invertible. It follows that the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
of the matrix must be equal to 0, so:
:
Solving for
, we have two positive solutions:
:
If we substitute ω
1 into the matrix and solve for (''A''
1, ''A''
2), we get (1, 1). If we substitute ω
2, we get (1, −1). (These vectors are
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s, and the frequencies are
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s.)
The first normal mode is:
:
Which corresponds to both masses moving in the same direction at the same time. This mode is called antisymmetric.
The second normal mode is:
:
This corresponds to the masses moving in the opposite directions, while the center of mass remains stationary. This mode is called symmetric.
The general solution is a
superposition of the normal modes where ''c''
1, ''c''
2, φ
1, and φ
2, are determined by the
initial conditions of the problem.
The process demonstrated here can be generalized and formulated using the formalism of
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
or
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momen ...
.
Standing waves
A
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
is a continuous form of normal mode. In a standing wave, all the space elements (i.e. (''x'', ''y'', ''z'') coordinates) are oscillating in the same
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
and in
phase (reaching the
equilibrium point together), but each has a different amplitude.
The general form of a standing wave is:
:
where ''ƒ''(''x'', ''y'', ''z'') represents the dependence of amplitude on location and the cosine\sine are the oscillations in time.
Physically, standing waves are formed by the
interference (superposition) of waves and their reflections (although one may also say the opposite; that a moving wave is a
superposition of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the ''ƒ''(''x'', ''y'', ''z'') form of the standing wave. This space-dependence is called a normal mode.
Usually, for problems with continuous dependence on (''x'', ''y'', ''z'') there is no single or finite number of normal modes, but there are infinitely many normal modes. If the problem is bounded (i.e. it is defined on a finite section of space) there are
countably many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
normal modes (usually numbered ''n'' = 1, 2, 3, ...). If the problem is not bounded, there is a continuous spectrum of normal modes.
Elastic solids
In any solid at any temperature, the primary particles (e.g. atoms or molecules) are not stationary, but rather vibrate about mean positions. In insulators the capacity of the solid to store thermal energy is due almost entirely to these vibrations. Many physical properties of the solid (e.g. modulus of elasticity) can be predicted given knowledge of the frequencies with which the particles vibrate. The simplest assumption (by Einstein) is that all the particles oscillate about their mean positions with the same natural frequency ''ν''. This is equivalent to the assumption that all atoms vibrate independently with a frequency ''ν''. Einstein also assumed that the allowed energy states of these oscillations are harmonics, or integral multiples of ''hν''. The spectrum of waveforms can be described mathematically using a Fourier series of sinusoidal density fluctuations (or thermal
phonons
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechan ...
).
Debye subsequently recognized that each oscillator is intimately coupled to its neighboring oscillators at all times. Thus, by replacing Einstein's identical uncoupled oscillators with the same number of coupled oscillators, Debye correlated the elastic vibrations of a one-dimensional solid with the number of mathematically special modes of vibration of a stretched string (see figure). The pure tone of lowest pitch or frequency is referred to as the fundamental and the multiples of that frequency are called its harmonic overtones. He assigned to one of the oscillators the frequency of the fundamental vibration of the whole block of solid. He assigned to the remaining oscillators the frequencies of the harmonics of that fundamental, with the highest of all these frequencies being limited by the motion of the smallest primary unit.
The normal modes of vibration of a crystal are in general superpositions of many overtones, each with an appropriate amplitude and phase. Longer wavelength (low frequency)
phonons
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechan ...
are exactly those acoustical vibrations which are considered in the theory of sound. Both longitudinal and transverse waves can be propagated through a solid, while, in general, only longitudinal waves are supported by fluids.
In the
longitudinal mode
A longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves confined in the cavity. The longitudinal modes correspond to the wavelengths of the wave which are reinforced by constructive interference after m ...
, the displacement of particles from their positions of equilibrium coincides with the propagation direction of the wave. Mechanical longitudinal waves have been also referred to as ''compression waves''. For
transverse mode
A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's propagation direction. Transverse modes occur in radio waves and microwa ...
s, individual particles move perpendicular to the propagation of the wave.
According to quantum theory, the mean energy of a normal vibrational mode of a crystalline solid with characteristic frequency ''ν'' is:
:
The term (1/2)''hν'' represents the "zero-point energy", or the energy which an oscillator will have at absolute zero. ''E''(''ν'') tends to the classic value ''kT'' at high temperatures
: